Question

M is a Riemannian manifold with $$Ric \ge - (n - 1)$$,f is a harmonic function on M,let $$h = |\nabla f|$$.By choosing an orthonormal basis such that $$|\nabla f|{e_1} = \nabla f,{e_\alpha }f = 0,\alpha \ne 1$$,we can get $$|{\nabla ^2}f{|^2} \ge {\frac{{n|\nabla |\nabla f||}}{{n - 1}}^2}$$.And by bochner formula$$\Delta {h^2} = 2Ri{c_M}(\nabla f,\nabla f) + 2|{\nabla ^2}f{|^2}$$,we finally get$$\Delta {h^2} \ge - 2(n - 1){h^2} + \frac{{2n}}{{(n - 1)}}|\nabla h{|^2}$$ If M is an Alexandrov space with$$ \sec \ge - 1$$,Given a function$$u \in W_{loc}^{1,2}\left( \Omega \right)$$we define a functional Δ on Lip0(Ω) by$${L_u} \left( \phi \right) = - \int_\Omega {\left\langle {\nabla u,\nabla \phi } \right\rangle } dvol,\forall \phi \in Li{p_0}\left( \Omega \right)$$ Suppose$$f \in W_{loc}^{1,2},{L_f}= 0$$$$|\nabla f| \ge c > 0$$for some constant c,we can get$$ \Delta {h^2} \ge - 2(n - 1){h^2} + \frac{{2n}}{{(n - 1)}}{\left( {\frac{{\left\langle {\nabla f,\nabla h} \right\rangle }}{h}} \right)^2}$$due to the paper"sharp spectral gap and Li-Yau's estimate on Alexandrov space",which is worse than in Riemannian case because$$ \frac{{\left\langle {\nabla f,\nabla h} \right\rangle }}{h} \ne |\nabla h|$$ So why does this happen?Can you give an Alexandrov space where The Bochner formula for Riemannian manifold is not valid?